In numerical estimates we will allow for \(\bar{g}_1^{NN}\)  as large as \(\mathcal {O}(10)\) , given that the \(\pi \pi \)  short-distance coupling \(\bar{g}_{LR}^{\pi \pi }=8.23\)  [1]} relevant for the vector–vector correlator by far exceeds its \(\mathcal {O}(1)\)  expectation.
However, the numerical impact of these poorly known short-distance parameters on the overall analysis is very minor,
as the main role of \({\cal A}_C^>\)  is to enforce the UV finiteness of the \(nn \rightarrow pp\)  amplitude.
